A 2000 kg elevator with broken cables is falling at 400 ms...

A 2000 kg elevator with broken cables is falling at 4.00 m/s when it first contacts a cushioning spring at the bottom of the shaft. The spring is supposed to eventually stop the elevator. The stiffness of the spring is 10.7 kN/m. During the motion a safety clamp applies a constant 16.5 kN frictional force to the elevator. 1) What is the speed of the elevator after it has moved downward 1.55 m from the point where it first contacts the spring including friction? 2) When the elevator is 1.55 m below point where it first contacts the spring, what is the magnitude of its acceleration? What is the direction of the acceleration here?

Transcript

00:01 Have a 2 ,000 kilogram elevator which is falling through a shaft and it hits a spring and get slowed down so i want to find this speed of elevator that move downwards 1 .55 meters so 0 .1 is going to be here or point 2 it's going to be wind as falling that distance to 1 .55 meters so our total energy 1 will be called our total energy a point 2 so our total energy point 1 is the kinetic energy which is half mv1 squared plus the potential energy a point 1 mg h1 this is the gravitational potential energy and you also add the elastic potential energy for the spring so that is going to be half k x squared this will be equal to total nj 0 1 over 2 mv2 squared plus m g h2 plus half k to x1 x2 squared we also need to add the walkdown by friction, which is the frictional force multiplied by the distance.

01:16 So you can substitute into this 1 over 2 times the mass, which is 2 ,000 times the velocity at 1 .1, which we're giving us 4 meters per second, 4 squared, plus 2 ,000 times 9 .8 .1 times the height, height, height 1, which is 1 .55, plus the spring constant key is given as 10 .7 kilo inches per meter.

01:47 10 ,700 newtons per meter multiplied by the extension of point one which is going to be zero because the spring is not yet compressed and plus 1 over 2 times 2 000 times v2 square now v2s we won't find which is the velocity of point 2 plus 2 000 times 9 .81 times height at point 2 which is going to be 0 because point 2 is going to be our reference point plus 1 over 2 times 10 ,700 times the extension of 0 .2 which is 1 .55 meters because that's how much the spring has compressed plus the walkdown by friction which is the friction of force 16 .5 kiloons that's 16 ,500 newtons multiplied by the distance through which it moves during that which is 1 .55 meters also so what's hoping for v2 this equation so this gives us v2 as 2 .83 meters per second approximately so for question 1 .2 one is the elevator sure when the elevator is 1 .5 meters below the point where it first contacts the sparing what is the magnitude of the acceleration and what is the direction of the acceleration so for that we just need to find the force first of all and the elastic force is equal to which is equal to the massum's acceleration.

03:36 So we're going to have the spring applying the elastic force upwards...

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